Batyrev's conjecture on the nef cone of curves

Let $(X,\Delta)$ be a projective KLT pair. Write $\overline{\operatorname{NE}}(X)_{K_X+\Delta>0}$ for the part of the closed cone of effective curves on which $K_X+\Delta$ is positive, and let $\operatorname{\overline{NM}}(X)$ denote the closed nef cone of curves. A curve is movable if it deforms in a family covering $X$. Batyrev's conjecture. There are countably many $(K_X+\Delta)$-negative movable curves $C_i$ such that

$$\overline{\operatorname{NE}}(X)_{K_X+\Delta>0}+\operatorname{\overline{NM}}(X)=\overline{\operatorname{NE}}(X)_{K_X+\Delta>0}+\sum\mathbb{R}_{>0}[C_i].$$

The rays $\mathbb{R}_{>0}[C_i]$ only accumulate along the hyperplane $(K_X+\Delta)^\bot$. The conjecture concerns the structure of the nef cone of curves and its relationship with the canonical divisor; the paper proves versions of it for threefolds in characteristic $p>5$ and in full generality in characteristic zero.